Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix}
2-\lambda & 1\\
3 & 4-\lambda
\end{bmatrix}\begin{bmatrix}
v_1\\
v_2
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$
$\begin{bmatrix}
2-\lambda & 1\\
3 & 4-\lambda
\end{bmatrix}=0$
$(\lambda -1) ( \lambda-5)=0$
$\lambda_1=1,\lambda_2=5$
2. Find eigenvectors:
For $\lambda=1$
let $B=A-\lambda_1I$
$B=\begin{bmatrix}
1 & 1\\
3 & 3
\end{bmatrix}$
Let $r$ be a free variable.
$\vec{V}=r(-1,1) \\
E_1=\{(-1,1)\}
\rightarrow dim(E_1)=1$
For $\lambda=5$
let $B=A-\lambda_1I$
$B=\begin{bmatrix}
-3 & 1\\
3 & -1
\end{bmatrix}$
Let $s$ be a free variable.
$\vec{V}=s(1,3) \\
E_2=\{(1,3)\}
\rightarrow dim(E_2)=1$
Hence, $A$ is non-defective.
